On the maximization of quadratic weighted kappa

An analytical expression for the optimal estimation of the numerical dependence by the criterion of a quadratic weighted kappa and also the expression for the optimal value of this criterion were obtained. It is shown that the optimal decision function is obtained from the regression function by a linear transformation. The coefficients of this transformation can be found from the condition of equality of mathematical expectations and variances of the predicted value and its estimate. The quadratic weighted kappa coefficient was originally proposed as an alternative to the correlation coefficient to reflect the strength of dependence between two characteristics, but recently it has been widely used as a criterion for the quality of the forecast in the problem of recovery of dependencies (regression analysis). At the same time, the properties of this coefficient in this context are still poorly understood. The properties of the quadratic weighted kappa criterion revealed in the work allow us to conclude that the expediency of using it as a criterion for the quality of the decision function in most cases raises doubts. This criterion provides a solution that is actually based on the regression function, but the variance of the forecast is artificially made equal to the variance of the original value. This distorts the forecast without improving the statistical properties of the decision function.